82 



Each curve of (lie moire-image corresponds to a certain constant 

 difference (or sum) of phase. 



The equation will first be derived for a constant difference of 

 phase 2(i — 2a' =z 2<9. 



This condition, togetlier with (1) and (II) gives: 



X — a 



z= cos 2 fp 



r 



X -\- a 



■=: COS 2 ff' 



{J 1 1) 



y — a 



2/ + « 



:=: COS 2 ff COS 2 ft — sin 2 ff sm 2 a 



= COS 2 {(f'—Oy-os 2 a—n?i 2 {q' — 6/) sm2ct 



{IV) 



Eliminating 2« from {IV) by means of the relation 

 sin' 2« -h cos* 2« = 1 

 we get : 



y — a 



cos 2 fp — si7i 2 (f 

 coê2{fp'-(9)-sin2{<p'-G) 



- Sl7l 2 (p 

 r 



y + « -Of' /A 

 S171 2 {<p f/) 



r 

 or after reduction : 



3 3 J/' 1 /ï' 



cos' ! 2 (rp-r//) + 2Ö I - 2 ^- COS |2 (.p-r/)') + 2 ^1 + 2 ^— -1=0 ( V) 



1' T 



When in this equation cos ip and cos cp' are replaced, resp. bj 



^^^^ and -^^, we get the equation of the moire-image in xy co- 

 r r 



ordi nates. 



It is, however, preferable to seek parameter equations. 



Suppose 2 {(p—<p') + 26» = 2A, then (V) becomes: 



r* COS* 2 A — 2 (?/*—«') cos 2 A + 2 {y* + a-) — ?•' = 

 which gives for y : 



y = ± V^r' cos^ A — a' cof^f" A . . . 

 The value of .i' follows from : 



2 (y— rp') + 2 6> = 2 A 

 cos 2 7) cos 2 r/:j' -}- sm 2 ff sin 2 r/)' = cos 2{L—6) 

 or, with regard to (III) and after reduction: 



r' cos' 2 (A-0 — 2 (,^■^ -a') cos 2 (A— <9) -h 2 (;?;' 4- a') — r' = 0. 



This equation, being of the same form as (VI), we get for x 



w = ± \/r' cos' {A— (9)— a* cotg* (A— <9). 



(VI) 

 {VII) 



